I'm worried that people are going to see the Ethnio screener over and over again...

Don't worry! Ethnio sets a cookie whenever a user views the screener. Unless it’s present on more than one domain, no user will see it twice. There are some issues with cookies and sub-domains, so it could be the case that if you have the same screener code on apples.ethnio.com and oranges.ethnio.com, a visitor could see the screener twice.